Question

1. from the fire tower in flatlands national park, a forest ranger sighted a fire. to measure the angle of depression of the fire, the ranger used an instrument that was known to be 32 meters above the ground. the angle of depression from the tower to the fire was 2°10. what was the distance between the fire and the base of the tower? tan2°10x = 810 -0378 = 32/x -0378x = 32

Explanation:

Step1: Convert angle to decimal degrees

First, convert $2^{\circ}10'$ to decimal - degrees. Since $1^{\circ}=60'$, then $10'=\frac{10}{60}\approx0.1667^{\circ}$. So $2^{\circ}10'\approx2 + 0.1667=2.1667^{\circ}$.

Step2: Set up tangent equation

Let the distance between the fire and the base of the tower be $x$. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the height of the instrument above the ground (opposite side) is 32 meters and the angle of depression $\theta = 2.1667^{\circ}$. So $\tan(2.1667^{\circ})=\frac{32}{x}$.

Step3: Solve for $x$

We can rewrite the equation as $x=\frac{32}{\tan(2.1667^{\circ})}$. Since $\tan(2.1667^{\circ})\approx0.0378$, then $x=\frac{32}{0.0378}\approx846.56$ meters.

Answer:

Approximately 846.56 meters